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   "cell_type": "markdown",
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   "source": [
    "# Navier-Stokes Solver using Finite Differences\n",
    "## CH EN 6355 - Computational Fluid Dynamics\n",
    "**Prof. Tony Saad (<a>www.tsaad.net</a>) <br/>Department of Chemical Engineering <br/>University of Utah**\n",
    "<hr/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "from matplotlib import cm\n",
    "plt.rcParams['animation.html'] = 'html5'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "'''\n",
    "Compute first derivative using central differencing on f\n",
    "'''\n",
    "def ddx(f, dx):\n",
    "    result = np.zeros_like(f)\n",
    "    result[1:-1,1:-1] = (f[1:-1,2:] - f[1:-1,:-2])/2.0/dx\n",
    "    return result\n",
    "\n",
    "def ddy(f, dy):\n",
    "    result = np.zeros_like(f)\n",
    "    result[1:-1,1:-1] = (f[2:,1:-1] - f[:-2,1:-1])/2.0/dy\n",
    "    return result\n",
    "    \n",
    "def laplacian(f, dx, dy):\n",
    "    result = np.zeros_like(f)\n",
    "    result[1:-1,1:-1] = (f[1:-1,2:] - 2.0*f[1:-1,1:-1] + f[1:-1,:-2])/dx/dx \\\n",
    "                      + (f[2:,1:-1] -2.0*f[1:-1,1:-1] + f[:-2,1:-1])/dy/dy\n",
    "    return result\n",
    "\n",
    "def div(u,v,dx,dy):\n",
    "    return ddx(u,dx) + ddy(v,dy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Re = 20.0\n"
     ]
    }
   ],
   "source": [
    "nx = 32\n",
    "ny = 32\n",
    "lx = 1.0\n",
    "ly = 1.0\n",
    "dx = lx/(nx-1)\n",
    "dy = ly/(ny-1)\n",
    "\n",
    "ν = 0.05\n",
    "Ut = 1.0 # m/s\n",
    "\n",
    "dt = min(0.25*dx*dx/ν, 4.0*ν/Ut/Ut)\n",
    "print('Re =', Ut*lx/ν)\n",
    "u = np.zeros([ny,nx])\n",
    "v = np.zeros([ny,nx])\n",
    "uh = np.zeros([ny,nx])\n",
    "vh = np.zeros([ny,nx])\n",
    "p = np.zeros([ny,nx])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pressure_poisson(p, dx, dy, b):\n",
    "    pn = np.empty_like(p)\n",
    "    it = 0\n",
    "    err = 1e5\n",
    "    tol = 1e-3\n",
    "    maxit = 50\n",
    "    while it < maxit and err > tol:\n",
    "        pn = p.copy()\n",
    "        p[1:-1, 1:-1] = (((pn[1:-1, 2:] + pn[1:-1, 0:-2]) * dy**2 + \n",
    "                          (pn[2:, 1:-1] + pn[0:-2, 1:-1]) * dx**2) /\n",
    "                          (2 * (dx**2 + dy**2)) -\n",
    "                          dx**2 * dy**2 / (2 * (dx**2 + dy**2)) * \n",
    "                          b[1:-1,1:-1])\n",
    "\n",
    "        p[:, -1] = p[:, -2] # dp/dy = 0 at right wall\n",
    "        p[0, :] = p[1, :]   # dp/dy = 0 at y = 0\n",
    "        p[:, 0] = p[:, 1]   # dp/dx = 0 at x = 0\n",
    "        p[-1, :] = 0        # p = 0 at the top wall\n",
    "        err = np.linalg.norm(p - pn, 2)\n",
    "        it += 1\n",
    "        \n",
    "    return p, err"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = 0.0\n",
    "tend = 4000*dt\n",
    "\n",
    "while t < tend:\n",
    "    # set boundary conditions\n",
    "    # bottom wall\n",
    "    u[0,:] = 0.0 \n",
    "    v[0,:] = 0.0     \n",
    "    # top wall    \n",
    "    u[-1,:] = Ut\n",
    "    v[-1,:] = 0.0\n",
    "    # left wall\n",
    "    u[:,0] = 0.0\n",
    "    v[:,0] = 0.0\n",
    "    # right wall\n",
    "    u[:,-1] = 0.0\n",
    "    v[:,-1] = 0.0\n",
    "        \n",
    "    # do the x-momentum RHS\n",
    "    # u rhs: - d(uu)/dx - d(vu)/dy + ν d2(u)\n",
    "    uRHS = - ddx(u*u,dx) - ddy(v*u,dy) + ν*laplacian(u,dx,dy)\n",
    "    # v rhs: - d(uv)/dx - d(vv)/dy + ν d2(v)\n",
    "    vRHS = - ddx(u*v,dx) - ddy(v*v,dy) + ν*laplacian(v,dx,dy)\n",
    "    \n",
    "    uh = u + dt*uRHS\n",
    "    vh = v + dt*vRHS\n",
    "    \n",
    "    # next compute the pressure RHS: prhs = div(un)/dt + div( [urhs, vrhs])\n",
    "    prhs = div(uh,vh,dx,dy)/dt\n",
    "    p,err = pressure_poisson(p,dx,dy,prhs)\n",
    "    \n",
    "    # finally compute the true velocities\n",
    "    # u_{n+1} = uh - dt*dpdx\n",
    "    u = uh - dt*ddx(p,dx)\n",
    "    v = vh - dt*ddy(p,dy)\n",
    "    t += dt"
   ]
  },
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   "source": [
    "vel =np.sqrt(uh**2 + vh**2)\n",
    "plt.contourf(vel,cmap='hsv')\n",
    "plt.quiver(u,v,color='k')   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
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   "source": [
    "divu = div(u,v,dx,dy)\n",
    "plt.imshow(abs(divu), origin='bottom')\n",
    "plt.colorbar()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x1187bd2e8>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
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     "metadata": {},
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   ],
   "source": [
    "divu = div(uh,vh,dx,dy) - dt * laplacian(p,dx,dy)\n",
    "plt.imshow(abs(divu), origin='bottom')\n",
    "plt.colorbar()"
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   "outputs": [
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       "CSS style adapted from https://github.com/barbagroup/CFDPython. Copyright (c) Barba group\n",
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       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
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       "\n",
       "/*  div#notebook {background-color: #1e1e1e; border-top: none;}\n",
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       "\n",
       "</style>\n",
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      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import urllib\n",
    "import requests\n",
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = requests.get(\"https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css\")\n",
    "    return HTML(styles.text)\n",
    "css_styling()\n",
    "\n"
   ]
  }
 ],
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